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Tai-Peng Tsai

Publications and source records attributed to Tai-Peng Tsai.

At least 19 recordsLinked to original sources
arXiv2026

Maximal regularity and caloric trace estimates in mixed Lebesgue norms for the heat equation

We study the heat equation in the half-space with nonhomogeneous Dirichlet boundary data. For the caloric extension $v$ of the boundary data $g$, we prove maximal regularity estimates in mixed Lebesgue norms $L^p_tL^q_x$ for any order derivative of $v$ in terms of mixed Besov and Lizorkin--Triebel type norms of $g$. We also establish the corresponding reverse inequalities, which are caloric trace estimates recovering the boundary regularity of $g$ from the mixed-norm regularity of $v$. As a model case, our results show that the natural \[\dotc W^{1,p}\big(\R;L^q(\R^d_+)\big)\cap L^p\big(\R;\dotc W^{2,q}(\R^d_+)\big)\] regularity norm of $v$ is controlled by the \[\dotc {F}^{1-\frac{1}{2q}}_{p,q}\big(\R;\,L^{q}(\R^{d-1})\big)\cap L^{p}\big(\R;\,\dotc{B}_{q,q}^{2-\frac{1}{q}}(\R^{d-1})\big)\] norm of $g$. The maximal regularity estimate holds for $1\leq p,q<\infty$, while the caloric trace estimate holds for $1<p<\infty$ and $1\leq q\leq\infty$. In particular, the endpoint cases $p=1$ or $q=1$ in the maximal regularity estimate are included and appear to be new. These endpoint estimates may be useful in the analysis of free-boundary Navier--Stokes problems with small initial data, whereas the caloric trace estimates may be relevant to the construction of Stokes or Navier--Stokes flows exhibiting strong boundary singularities.

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arXiv2026

On the regularity of axisymmetric, swirl-free solutions of the Euler equation in four and higher dimensions

In this paper, we consider axisymmetric, swirl-free solutions of the Euler equation in four and higher dimensions. We show that in dimension $d\geq 4$, axisymmetric, swirl-free solutions of the Euler equation have properties which could allow finite-time singularity formation of a form that is excluded when $d=3$, and we prove a conditional blowup result for axisymmetric, swirl-free solutions of the Euler equation in dimension $d\geq 4$. The condition which must be imposed on the solution in order to imply blowup becomes weaker as $d\to +\infty$, suggesting the dynamics are becoming much more singular as the dimension increases.

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arXiv2026

Existence and regularity for perturbed Stokes system with critical drift in 2D

We consider a perturbed Stokes system with critical divergence-free drift in a bounded Lipschitz domain in $R^2$, with sufficiently small Lipschitz constant L. It extends our previous work in $\Bbb R^n, n\ge 3$, to two-dimensional case. For large drift in weak $L^2$ space, we prove unique existence of q-weak solutions for force in $L^q$ with q close to 2. Moreover, for drift in $L^2(\Bbb R^2)$ we prove the unique existence of $W^{1,2}$ solutions for arbitrarily large L. Using similar methods we can also prove analogous results for scalar equations with divergence-free drifts in weak $L^2$ space.

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arXiv2026

Existence and regularity for perturbed Stokes system with critical drift

We consider the existence and $L^q$ gradient estimates for perturbed Stokes systems with divergence-free critical drift in a bounded Lipschitz domain in $\mathbb{R}^n$, $n \ge 3$. The first two results assume the drift is either in $L^n$ or sufficiently small in weak $L^n$. The third result assumes the drift is in weak $L^n$ without smallness, and obtain results for $q$ close to 2.

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arXiv2025

Analysis of quasi-periodic waves of cubic nonlinear Schr{ö}dinger equations

We study the quasi-periodic standing wave solutions of the focusing and defocusing cubic nonlinear Schr{ö}dinger equations in dimension one. In the defocusing case, we establish a diffeomorphic correspondence between the invariants of the ordinary differential equation of the wave profiles and the conserved quantities of the evolution equation. We introduce a numerical scheme to compute the minimizers of the energy at fixed mass and momentum for both focusing and defocusing cases. The scheme is based on a gradient flow approach with discrete renormalization at each time step. The novelty of our scheme is that the renormalization step deals at the same time with the mass and the momentum constraints. In numerical experiments, we observe that a given solution of the profile ordinary differential equation is also a minimizer of the energy at corresponding mass and momentum.

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arXiv2025

The local regularity theory for the Stokes and Navier--Stokes equations near the curved boundary

In this paper, we study local regularity of the solutions to the Stokes equations near a curved boundary under no-slip or Navier boundary conditions. We extend previous boundary estimates near a flat boundary to that near a curved boundary, under very low starting regularity assumptions. Compared with the flat case, the proof for the curved case is more complicated and we adapt new techniques such as the ``normal form" after the mollification, recovering vertical derivative estimates from horizontal derivative estimates, and transferring temporal derivatives to spatial derivatives, to deal with the higher order perturbation terms generated by boundary straightening. As an application, we propose a new definition of boundary regular points for the incompressible Navier--Stokes equations that guarantees higher spatial regularity.

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arXiv2024

Large discretely self-similar solutions to Oberbeck-Boussinesq system with Newtonian gravitational field

Discretely self-similar solutions to Oberbeck-Boussinesq system with Newtonian gravitational field for large discretely self-similar initial data are constructed in this note, extending the construction of Brandolese and Karch (arXiv:2311.01093) on self-similar solutions. It follows the approach of Bradshaw and Tsai (Ann.~Henri Poincaré 2017) and find an explicit a priori bound for the deviation from suitably revised profiles in similarity variables.

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arXiv2024

Applications of the Green tensor estimates of the nonstationary Stokes system in the half space

In this paper, we present a series of applications of the pointwise estimates of the (unrestricted) Green tensor of the nonstationary Stokes system in the half space, established in our previous work [CMP 2023]. First, we show the $L^1$-$L^q$ estimates for the Stokes flow with possibly non-solenoidal $L^1$ initial data, generalizing the results of Giga-Matsui-Shimizu [Math. Z. 1999] and Desch-Hieber-Prüss [J. Evol. Equ. 2001]. Second, we construct mild solutions of the Navier-Stokes equations in the half space with mixed-type pointwise decay or with pointwise decay alongside boundary vanishing. Finally, we explore various coupled fluid systems in the half space including viscous resistive magnetohydrodynamics equations, a coupled system for the flow and the magnetic field of MHD type, and the nematic liquid crystal flow. For each of these systems, we construct mild solutions in $L^q$, pointwise decay, and uniformly local $L^q$ spaces.

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arXiv2024

Poisson kernel and blow-up of the second derivatives near the boundary for Stokes equations with Navier boundary condition

We derive the explicit Poisson kernel of Stokes equations in the half space with nonhomogeneous Navier boundary condition (BC) for both infinite and finite slip length. By using this kernel, for any $q>1$, we construct a finite energy solution of Stokes equations with Navier BC in the half space, with bounded velocity and velocity gradient, but having unbounded second derivatives in $L^q$ locally near the boundary. While the Caccioppoli type inequality of Stokes equations with Navier BC is true for the first derivatives of velocity, which is proved by us in [CPAA 2023], this example shows that the corresponding inequality for the second derivatives of the velocity is not true. Moreover, we give an alternative proof of the blow-up using a shear flow example, which is simple and is the solution of both Stokes and Navier--Stokes equations.

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arXiv2023

Global Navier-Stokes flows in intermediate spaces

We construct global weak solutions of the three dimensional incompressible Navier-Stokes equations in intermediate spaces between the space of uniformly locally square integrable functions and Herz-type spaces which involve weighted integrals centered at the origin. Our results bridge the existence theorems of Lemarié-Rieusset and of Bradshaw, Kukavica and Tsai. An application to eventual regularity is included which generalizes the prior work of Bradshaw, Kukavica and Tsai as well as Bradshaw, Kukavica and Ozanski.

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arXiv2023

On linear elliptic equations with drift terms in critical weak spaces

We study the Dirichlet problem for a second order linear elliptic equation in a bounded smooth domain $Ω$ in $\mathbb{R}^n$, $n \ge 3$, with the drift $\mathbf{b} $ belonging to the critical weak space $L^{n,\infty}(Ω)$. We decompose the drift $\mathbf{b} = \mathbf{b}_1 + \mathbf{b}_2$ in which $\text{div} \mathbf{b}_1 \geq 0$ and $\mathbf{b}_2$ is small only in a small scale quasi-norm of $L^{n,\infty}(Ω)$. Under this new smallness condition, we prove existence, uniqueness, and regularity estimates of weak solutions to the problem and its dual. Hölder regularity and derivative estimates of weak solutions to the dual problem are also established. As a result, we prove uniqueness of very weak solutions slightly below the threshold. When $\mathbf{b}_2 =0$, our results recover those by Kim and Tsai in [SIAM J. Math. Anal. 52 (2020)]. Due to the new small scale quasi-norm, our results are new even when $\mathbf{b}_1=0$.

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arXiv2023

Gradient estimates for the non-stationary Stokes system with the Navier boundary condition

For the non-stationary Stokes system, it is well-known that one can improve spatial regularity in the interior, but not near the boundary if it is coupled with the no-slip boundary condition. In this note we show that, to the contrary, spatial regularity can be improved near a flat boundary if it is coupled with the Navier boundary condition, with either infinite or finite slip length. The case with finite slip length is more difficult than the case with infinite slip length.

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arXiv2023

Growth rates for anti-parallel vortex tube Euler flows in three and higher dimensions

We consider axisymmetric, swirl-free solutions of the Euler equations in three and higher dimensions, of generalized anti-parallel-vortex-tube-pair-type: the initial scalar vorticity has a sign in the half-space, is odd under reflection across the plane, is bounded and decays sufficiently rapidly at the axis and at spatial infinity. We prove lower bounds on the growth of such solutions in all dimensions, improving a lower bound proved by Choi and Jeong arXiv:2110.09079 in three dimensions.

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arXiv2022

The Green tensor of the nonstationary Stokes system in the half space

We prove the first ever pointwise estimates of the (unrestricted) Green tensor and the associated pressure tensor of the nonstationary Stokes system in the half-space, for every space dimension greater than one. The force field is not necessarily assumed to be solenoidal. The key is to find a suitable Green tensor formula which maximizes the tangential decay, showing in particular the integrability of Green tensor derivatives. With its pointwise estimates, we show the symmetry of the Green tensor, which in turn improves pointwise estimates. We also study how the solutions converge to the initial data, and the (infinitely many) restricted Green tensors acting on solenoidal vector fields. As applications, we give new proofs of existence of mild solutions of the Navier-Stokes equations in $L^q$, pointwise decay, and uniformly local $L^q$ spaces in the half-space.

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arXiv2022

Mild solutions and spacetime integral bounds for Stokes and Navier-Stokes flows in Wiener amalgam spaces

We first prove decay estimates and spacetime integral bounds for Stokes flows in amalgam spaces $E^r_q$ which connect the classical Lebesgue spaces to the spaces of uniformly locally $r$-integrable functions. Using these estimates, we construct mild solutions of the Navier-Stokes equations in the amalgam spaces satisfying the corresponding spacetime integral bounds. Time-global solutions are constructed for small data in $E^3_q$, $1\le q \le 3$. Our results provide new bounds for the strong solutions classically constructed by Kato and the more recent solutions in uniformly local spaces constructed by Maekawa and Terasawa. As an application we obtain a result on the stability of suitability for weak solutions to the perturbed Navier-Stokes equation where the drift velocity solves the Navier-Stokes equations and has small data in a local $L^3$ class. Extending an earlier result, we also construct global-in-time local energy weak solutions in $E^2_q$, $1\le q <2$.

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